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‘Modern’ differential geometry in elementary particle physics

Gauge fields, solitons, superspaces, quantum differential geometry, etc.
  • Robert Hermann
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 77)

Keywords

Gauge Transformation Differential Form Gauge Field Nonlinear Evolution Equation Parallel Transport 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. 1.
    R. Hermann, Lie Groups for Physicists, W.A. Benjamin, Inc., New York, 1966.Google Scholar
  2. 2.
    R. Hermann, Differential Geometry and the Calculus of Variations, Academic Press, New York, 1969. Second edition to be published by Math Sci Press, Brookline, Mass.Google Scholar
  3. 3.
    R. Hermann, Fourier Analysis on Groups and Partial Wave Analysis, W.A. Benjamin, New York, 1969.Google Scholar
  4. 4.
    R. Hermann, Lie Algebras and Quantum Mechanics, W.A. Benjamin, New York, 1970.Google Scholar
  5. 5.
    R. Hermann, Vector Bundles in Mathematical Physics, Parts I and II, W.A. Benjamin, New York, 1970.Google Scholar
  6. 6.
    R. Hermann, Lectures on Mathematical Physics, Vol. 1, W.A. Benjamin, New York, 1970.Google Scholar
  7. 7.
    R. Hermann, Lectures on Mathematical Physics, Vol. II, W.A. Benjamin, Reading, Mass, 1972.Google Scholar
  8. 8.
    R. Hermann, Geometry, Physics and Systems, Marcel Dekker, New York, 1973.Google Scholar
  9. 9.
    R. Hermann, Physical Aspects of Lie Group Theory, University of Montreal Press, Montreal, 1974.Google Scholar
  10. 10.
    R. Hermann, Energy-Momentum Tensors, Vol. IV of Interdisciplinary Mathematics, Math Sci Press, Brookline, Mass., 1973.Google Scholar
  11. 11.
    R. Hermann, Topics in General Relativity, Vol. V of Interdisciplinary Mathematics, Math Sci Press, Brookline, Mass., 1973.Google Scholar
  12. 12.
    R. Hermann, Topics in the Mathematics of Quantum Mechanics, Vol. VI of Interdisciplinary Mathematics, Math Sci Press, Brookline, Mass., 1973.Google Scholar
  13. 13.
    R. Hermann, Geometric Structure Theory of Systems-Control Theory and Physics, Part A, Vol. IX of Interdisciplinary Mathematics, Math Sci Press, Brookline, Mass.Google Scholar
  14. 14.
    R. Hermann, Gauge Fields and Cartan-Ehresmann Connections, Part A, Vol. X of Interdisciplinary Mathematics, Math.Sci Press, Brookline, Mass., 1975.Google Scholar
  15. 15.
    R. Hermann, Geometric Structure of Systems-Control Theory and Physics, Part B, Vol. XI of Interdisciplinary Mathematics, Math Sci Press, Brookline, Mass., 1976.Google Scholar
  16. 16.
    R. Hermann, Geometry of Non-Linear Differential Equations, Bäcklund Transformations and Solitons, Parts A and B, Vols. XII and XIV of Interdisciplinary Mathematics, Math Sci Press, Brookline, Mass., 1976, 1977.Google Scholar
  17. 16a.
    R. Hermann, Toda Lattices, Cosymplectic Manifolds, Bäcklund Transformations and Kinks, Part A, Interdisciplinary Mathematics, Vol. XV, Math Sci Press, Brookline, Mass., 1977.Google Scholar
  18. 17.
    W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York, 1975.Google Scholar
  19. 18.
    C. Misner, K. Thorne and J. Wheeler, Gravitation, W.H. Freeman, San Francisco, 1973.Google Scholar
  20. 19.
    A. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962.Google Scholar
  21. 20.
    S. Kubayashi and K. Nomizu, Foundations of Differential Geometry, Interscience, New York, 1973.Google Scholar
  22. 21.
    R. Bishop and S. Goldberg, Tensor Analysis on Manifolds, Macmillan, New York, 1968.Google Scholar
  23. 22.
    E. Cartan, Oeuvres completes, Gauthier-Villar, Paris, 1952.Google Scholar
  24. 23.
    C. Chevalley, Lie Groups, Princeton Univ. Press, 1946.Google Scholar
  25. 24.
    G. Darboux, Théorie genérale des surfaces, Chelsea, New York.Google Scholar
  26. 25.
    A. Forsyth, Theory of Differential Equations, Dover, New York.Google Scholar
  27. 26.
    A. Kumpera and D.C. Spencer, Lie Equations, Princeton Univ. Press, 1972.Google Scholar
  28. 27.
    J.C. Taylor, Gauge Invariance of Weak Interaction, Cambridge Univ. Press, 1976.Google Scholar
  29. 28.
    H.D. Wahlquist and F.B. Estabrook, Prolongation structures of nonlinear evolution equations, J. Math. Phys. 16 (1975), 1–7.Google Scholar
  30. 28a.
    Sophus Lie's 1880 Transformation Group Paper, comments and additional material by R. Hermann (Lie Groups: History, Frontiers and Applications, Vol. 1) Math Sci Press, Brookline, Mass., 1975.Google Scholar
  31. 28b.
    Ricci and Levi-Civita's Tensor Analysis Paper, translation, comments and additional material by R. Hermann (Lie Groups: History, Frontiers and Applications, Vol. 2) Math Sci Press, 1975.Google Scholar
  32. 28c.
    Sophus Lie's 1884 Differential Invariant Paper, comments and additional material by R. Hermann (Lie Groups: History, Frontiers and Applications, Vol. 3) Math Sci Press, 1975.Google Scholar
  33. 29.
    H. Cartan, Notions d'algebre, différentielles, Colloque de topologie de Bruxelles, Masson and Cie, Paris, 1950.Google Scholar
  34. 30.
    R. Hermann, Quantum and Fermion Differential Geometry, Part A, Math Sci Press, Brookline, Mass. 1950.Google Scholar
  35. 31.
    F. Mansouri, Differential geometry in graded manifolds, J. Math. Phys. 18 (1977) 52.Google Scholar
  36. 32.
    R. Casabuoni, Nuovo Cim 33A (1976), 389.Google Scholar
  37. 33.
    F. Berezin and G.I. Kac, Mat. Sb. USSR, 82 (1970), 124; Eng. translation 11 (1970), 311.Google Scholar
  38. 34.
    J. Wess and B. Zumino, Nucl. Phys. B70 (1974), 39.Google Scholar
  39. 35.
    L. Corwin, Y. Ne'eman and S. Sternberg, Reviews of Modern Physics 47 (1975), 573.Google Scholar
  40. 36.
    B. Zumino, in Gauge Theories and Modern Field Theory, ed. by R. Arnowitt and P. Natte, MIT Press, 1976.Google Scholar
  41. 37.
    B. Kostant, Graded manifolds, graded Lie theory and prequantization, preprint, MIT Math. Dept.Google Scholar
  42. 38.
    F. Berezin and M. Marinov, Particle spin dynamics as the Grassmann variant of classical mechanics, preprint, Moscow, 1976.Google Scholar
  43. 39.
    E. Cartan, Lecons sur les invariants integraux, Hermann et Cie, Paris, 1971.Google Scholar
  44. 40.
    E. Cartan, Les systémes différentielles exterieures et leurs applications géométriques Google Scholar
  45. 41.
    W.V.D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge Press, 1941.Google Scholar
  46. 42.
    G. de Rham, Varietés différentielles, Hermann, Paris, 1955.Google Scholar
  47. 43.
    B.K. Harrison and F.B. Estabrook, Geometric appraoch to invariance groups and solution of partial differential systems, J. Math. Phys. 12 (1971) 653–666.Google Scholar
  48. 44.
    H.D. Wahlquist and F.B. Estabrook, Bäcklund transformation for solution of the Korteweg-de Vries equation, Phys. Rev. Lett. 31 (1973), 1386–1390.Google Scholar
  49. 45.
    F. Estabrook, Comments on generalized Hamiltonian dynamics, Phys. Rev. D8 (1973), 2740–2743.Google Scholar
  50. 46.
    F.B. Estabrook, Some old and new techniques for the practical use of exterior differential forms, in Robert M. Miura, ed., Bäcklund Transformation, the Inverse Scattering Method. Solitons and Their Application, Lecture Notes in Mathematics, No. 515, Springer-Verlag, Berlin, New York, 1976.Google Scholar
  51. 47.
    H.D. Wahlquist, Bäcklund transformation of potentials of the Korteweg-de Vries equation and the interaction of solitons with cnoidal waves, in Robert M. Miura, ed., Bäcklund Transformation, the Inverse Scattering Method. Solitons and Their Application, Lecture Notes in Mathematics, No. 515, Springer-Verlag, Berlin, New York, 1976.Google Scholar
  52. 48.
    J. Corones and F.J. Testa, Pseudopotentials and their applications, in Robert M. Miura, ed., Bäcklund Transformation, the Inverse Scattering Method. Solitons and Their Application, Lecture Notes in Mathematics, No. 515, Springer-Verlag, Berlin, New York, 1976.Google Scholar
  53. 49.
    F.B. Estabrook and H.D. Wahlquist, The geometric approach to sets of ordinary differential equations and Hamiltonian Dynamics, SIAM Rev. 17 (1975), 201–220.Google Scholar
  54. 50.
    R. Hermann, The pseudopotentials of Estabrook and Wahlquist, the geometry of solitons, and the theory of connections, Phys. Rev. Lett. 36 (1976), 835.Google Scholar
  55. 51.
    H.C. Morris, Prolongation structures and a generalized inverse scattering problem, J. Math. Phys. 17 (1976), 1867–1869.Google Scholar
  56. 52.
    J. Corones, Solitons and simple pseudopotentials, J. Math. Phys. 17 (1976), 756–759.Google Scholar
  57. 53.
    F.B. Estabrook and H.D. Wahlquist, Prolongation structures of nonlinear evolution equations, II, J. Math. Phys. 17 (1976) 1293–1297.Google Scholar
  58. 54.
    C. Morris, Prolongation structures and nonlinear evolution equations in two spatial dimensions, J. Math. Phys. 17 (1976), 1870–1872.Google Scholar
  59. 55.
    H.C. Morris, Prolongation structures and nonlinear evolution equations in two spatial dimensions, II: A generalized nonlinear Schrödinger equation, J. Math. Phys. 18 (1977), 285–288.Google Scholar
  60. 56.
    B.K. Harrison, Remarks on the problem of two neighboring black holes, Utah Academy Proceedings 53 (1976), 67–74.Google Scholar
  61. 57.
    H.C. Morris, A prolongation structure for the AKNS system and its generalization, J. Math. Phys. 18 (1977), 533–536.Google Scholar
  62. 58.
    J.C. Corones, Solitons, pseudopotentials and certain Lie algebras, J. Math. Phys. 18 (1977), 163–164.Google Scholar
  63. 59.
    H.C. Morris, Prolongation structures and nonlinear evolution equations in two spatial dimensions, III: A general class of equations, TCD 1976-7 (submitted to J. Math. Phys., 1976).Google Scholar
  64. 60.
    R.K. Dodd and J.D. Gibbon, The prolongation structure of some higher order Korteweg-de Vries equations (preprint, 1977).Google Scholar
  65. 61.
    H.C. Morris, A generalized prolongation structure and the Bäcklund transformation of the anticommuting massive thirring model, TCD-1977-2 (preprint, 1977).Google Scholar
  66. 62.
    R.K. Dodd and J.D. Gibbon:, The prolongation structure of a class of nonlinear evolution equations (preprint, 1977).Google Scholar
  67. 63.
    M. Crampin, F.A.E. Pirani and D.C. Robinson, The soliton connection (preprint, 1977).Google Scholar
  68. 64.
    H.C. Morris, Prolongation structure of nonlinear evolution equations in two and three dimensiona, Seminar/Institute on Differential and Algebraic Geometry for Control Engineers, NASA 1976.Google Scholar
  69. 65.
    H.C. Morris, Soliton solutions and the higher order Korteweg-de Vries equations, J. Math. Phys. 18 (1977), 530–532.Google Scholar
  70. 66.
    R. Hermann, bd., Proceedings of the Ames (NASA) 1976 Conference on Geometric Non-Linear Waves. Articles by F. Estabrook, R. Hermann, H. Wahlquist, J. Corones, H. Morris, R. Gardner, A. Scott; MATH SCI PRESS, Brookline, Mass., 1977.Google Scholar
  71. 67.
    M. Wadati, H. Sanuki and K. Konno, Prog. Theor. Phys. 53 (1975), 419.Google Scholar
  72. 68.
    K. Konno and M. Wadati, Prog. Theor. Phys. 53 (1975), 1652.Google Scholar
  73. 69.
    D. Finkelstein, Kinks, J. Math. Phys. 7 (1966), 1218–1228.Google Scholar
  74. 70.
    D. Finkelstein and D. Rubenstein, J. Math. Phys. 9 (1968), 1762.Google Scholar
  75. 71.
    E. Goursat, Lecons sur le probléme de Pfaff, Hermann, Paris, 1972.Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Robert Hermann
    • 1
  1. 1.Department of Physics Lyman LaboratoryHarvard UniversityCambridge

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